The Associative Property of Addition

πŸ†Practice associative property

The associative property of addition allows us to group two addends and then add the third addend to the result.


We can use this property in three ways:
1. We start by adding the first and the second addend, solve the sum and add the third addend to the result.
2. We start by adding the second and third addends, solve the sum and then add the first addend to the result.

  1. We start by adding the first and the third addend, solve the sum and then add the second addend to the result.


We will place in parentheses around the addends that we want to group first to give them priority in the order of operations.

The associative property of addition also works in algebraic expressions, but not in subtraction operations.
Let's define the associative property of addition as:
a+b+c=(a+b)+c=a+(b+c)=(a+c)+ba+b+c=(a+b)+c=a+(b+c)=(a+c)+b

A - The Associative Property of Addition

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Test yourself on associative property!

einstein

\( 94+12+6= \)

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Let's look at an example:
2+10X+2X=2+12X2+10X+2X=2+12X

We can group the second and third addends, add them, and then add the first addend to the result.
We will obtain an expression equivalent to the first expression since the associative property does not change the result.


Let's put a number in place of X to verify :

X=3X=3
2+10Γ—3+2Γ—3=2+12Γ—3 2+10\times 3+2\times 3=2+12\times 3

2+30+6=2+362+30+6=2+36
38=3838=38

As we can see, after applying the associative property and adding the last two terms of the expression(the second and the third), and then adding the first term to the sum, the result is the same.
The associative property is an important and useful tool, so we recommend that you practice as much as possible until you can use it without thinking!


Example exercises

Exercise 1

Task:

7+8+12= 7+8+12=

Solution:

7+8+12=15+12=27 7+8+12=15+12=27

Answer:

27 27


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Exercise 2

Task:

94+12+6= 94+12+6=

Solution:

94+6+12=100+12=112 94+6+12=100+12=112

Answer:

112 112


Exercise 3

Task:

13+37+45= 13+37+45=

50+45=95 50+45=95

Answer:

95 95


Do you know what the answer is?

Exercise 4

Task:

4:2Γ—(5+4+6)= 4:2Γ—\left(5+4+6\right)=

Solution:

2Γ—(5+4+6)= 2Γ—\left(5+4+6\right)=

2Γ—(5+4+6)= 2\times(5+4+6)=

2Γ—(15)= 2\times(15)=

2Γ—(15)=30 2\times(15)=30

Answer:

30 30


Exercise 5

Task:

723+616+419=? 7\frac{2}{3}+6\frac{1}{6}+4\frac{1}{9}=\text{?}

Solution:

723+616+419= 7\frac{2}{3}+6\frac{1}{6}+4\frac{1}{9}=

=71218+6318+4218=17+12+3+218 =7\frac{12}{18}+6\frac{3}{18}+4\frac{2}{18}=17+\frac{12+3+2}{18}

=71718=171718 =7\frac{17}{18}=17\frac{17}{18}

Answer:

171718 17\frac{17}{18}


Check your understanding

Exercise 6

Task:

6Γ—45x+7Γ—12x+4Γ—13xΓ—7.3x=?6\times\frac{4}{5}x+7\times\frac{1}{2}x+4\times\frac{1}{3}x\times7.3x=\text{?}

Solution:

645X+712X+413XΓ—7310X=345X+152X+133XΓ—7310X6\frac{4}{5}X+7\frac{1}{2}X+4\frac{1}{3}X\times7\frac{3}{10}X=\frac{34}{5}X+\frac{15}{2}X+\frac{13}{3}X\times\frac{73}{10}X

=6810X+7510X+13Γ—733Γ—10X2 =\frac{68}{10}X+\frac{75}{10}X+\frac{13\times73}{3\times10}XΒ²

=68+7510X+94930X2 =\frac{68+75}{10}X+\frac{949}{30}XΒ²

=14310X+311930X2=14.3X+311930X2=\frac{143}{10}X+31\frac{19}{30}XΒ²=14.3X+31\frac{19}{30}XΒ²

Answer:

14.3x+311930x2 14.3x+31\frac{19}{30}xΒ²


Exercise 7

Task:

4.1Γ—1.6Γ—3.2+4.7=? 4.1\times1.6\times3.2+4.7=\text{?}

Solution:

4.1Γ—1.6Γ—3.2+4.7=? 4.1\times1.6\times3.2+4.7=\text{?}

=4110Γ—1610Γ—3210+4710==4\frac{1}{10}\times1\frac{6}{10}\times3\frac{2}{10}+4\frac{7}{10}=

4110Γ—1610Γ—3210+4710=656100Γ—3210+4710 \frac{41}{10}\times\frac{16}{10}\times\frac{32}{10}+\frac{47}{10}=\frac{656}{100}\times\frac{32}{10}+\frac{47}{10}

41Γ—1610Γ—10=656100=656Γ—32100Γ—10+4710=\frac{41\times16}{10\times10}=\frac{656}{100}=\frac{656\times32}{100\times10}+\frac{47}{10}=

=209921000+47001000 =\frac{20992}{1000}+\frac{4700}{1000}

=256921000=25.692 =\frac{25692}{1000}=25.692

Answer:

25.692 25.692


Do you think you will be able to solve it?

examples with solutions for the associative property of addition

Exercise #1

6:2+9βˆ’4= 6:2+9-4=

Video Solution

Step-by-Step Solution

According to the order of operations, we first solve the division exercise, and then the subtraction:

(6:2)+9βˆ’4= (6:2)+9-4=

6:2=3 6:2=3

Now we place the subtraction exercise in parentheses:

3+(9βˆ’4)= 3+(9-4)=

3+5=8 3+5=8

Answer

8 8

Exercise #2

3Γ—5Γ—4= 3\times5\times4=

Video Solution

Step-by-Step Solution

According to the order of operations, we must solve the exercise from left to right.

But, this can leave us with awkward or complicated numbers to calculate.

Since the entire exercise is a multiplication, you can use the associative property to reorganize the exercise:

3*5*4=

We will start by calculating the second exercise, so we will mark it with parentheses:

3*(5*4)=

3*(20)=

Now, we can easily solve the rest of the exercise:

3*20=60

Answer

60

Exercise #3

3+2βˆ’11= 3+2-11=

Video Solution

Step-by-Step Solution

According to the order of operations, we solve the exercise from left to right:

3+2=5 3+2=5

5βˆ’11=βˆ’6 5-11=-6

Answer

βˆ’6 -6

Exercise #4

4+5+1βˆ’3= 4+5+1-3=

Video Solution

Step-by-Step Solution

According to the order of operations, we solve the exercise from left to right:

4+5=9 4+5=9

9+1=10 9+1=10

10βˆ’3=7 10-3=7

Answer

7

Exercise #5

24:8:3= 24:8:3=

Video Solution

Step-by-Step Solution

According to the order of operations, we solve the exercise from left to right since the only operation in the exercise is division:

24:8=3 24:8=3

3:3=1 3:3=1

Answer

1 1

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